Revista de la
Unión Matemática Argentina
Families of convex tilings
Richard Kenyon
Volume 64, no. 1 (2022), pp. 87–101

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We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile “orientations” determine the tiling. We associate to a tiling an underlying bipartite planar graph $\mathcal{G}$ and its corresponding Kasteleyn matrix $K$. If $\mathcal{G}$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.